Question

If cos α = 2/3, then range of the values of φ for which the point φ on the ellipse x² + 4y² = 4 falls inside the circle x² + y² + 4x + 3 = 0 is:

• A. (-α, α)
• B. (0, α)
• C. (α, π)
• D. (π - α, π + α)

Answer 1

Answer: (D)

(π - α, π + α)

Answer 2

The point φ on the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ will be

(2 cosφ, sinφ). It lies inside the circle

x² + y² + 4x + 3 = 0, if

4cos² φ + sin² φ + 8 cosφ + 3 < 0

⇒ 3cos²φ + 8 cosφ + 4 < 0

⇒ (cosφ + 2) (3 cosφ + 2) < 0

⇒ -2 < cosφ < - 2/3

⇒ - 1≤ cosφ < cos (π - α)

∴ π - α < φ < π + α is most appropriate.